Convergence for a planar elliptic problem with large exponent Neumann data

2019/11/29 by Habib Fourti, Fourti, Habib
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP

paper · pdf · doi:10.48550/arxiv.1912.01453

arXiv admin note: text overlap with arXiv:1602.06919 by other authors

arxiv created 2019/11/29 · openalex publication_date 2019/11/29 · arxiv updated 2019/12/04 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We study positive solutions u_p of the nonlinear Neumann elliptic problem Δu =u in Ω, ∂ u/∂ν= |u|^p-1u on ∂Ω, where Ω is a bounded open smooth domain in ℝ². We investigate the asymptotic behavior of families of solutions u_p satisfying an energy bound condition when the exponent p is getting large. Inspired by the work of Davila-del Pino-Musso \citeDavilaDM, we prove that u_p is developing m peaks x_i∈∂ Ω, in the sense u_p^p/∫_{∂ Ω}u_p^p approaches the sum of m Dirac masses at the boundary and we determine the localization of these concentration points.

Citations

• Asymptotic analysis for the Lane-Emden problem in dimension two

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